Assume a, b and c are positive integers.
1) Suppose that a | b. Show that gcd(a, c) ≤ gcd(b, c).
2) Suppose that a ≤ b. Is it necessarily true that gcd(a, c) ≤ gcd(b, c)?
I'm having trouble with question number 2). My answer for 1 is since a divides b, there exists an integer k such that ak=b. By substituting in ak for b, gcd(a, c) <= gcd(ak, c). Since k is an integer that means the statement is true.
I don't really know how to go about question 2, can anybody explain why it is true or isn't? Thanks! Also if there is anything wrong with my solution for 1, please let me know. Thanks!
Let $a=2, b=3, c=4$. Then $$\gcd(a,c)=2>1=\gcd(b,c)$$